The epidemic threshold \(\tau \)  is defined as follows [1]}:
\(\boxed{{\left\lbrace \begin{array}{ll}\mathcal {R}_0 =\dfrac{\beta }{\delta } < \tau (A) \Rightarrow \text{infection dies out over time}\\\mathcal {R}_0 =\dfrac{\beta }{\delta } > \tau (A) \Rightarrow \text{infection grows over time}\\\end{array}\right.}}\) 
